Cassandra wasn't really close to the other Seekers even if she considered them her family, she mentions it in a conversation.
Yes, I didn't mean to imply that becoming a Seeker meant she was surrounded by close friends all of a sudden -- just that before then, she probably never had much of a chance to interact with people as peers at all. And even after she became a Seeker, she likely spent most of her time away from whatever friends she made there, doing Seeker-work in the field. Her whole life has been one that does not really enable forming close ties.
As for her reading people skills, she is actually suspicious of most companions.
At least initially, yes, but she does open up. Her relationship with Cole is the best example for how far she can come to see the good in people. It is, as I said, a shame that the changes in the way two characters feel about each other are rarely reflected anywhere but in the banters between them. For example, Cass' randomly selected lines when you ask her about the others in Skyhold never change. As a result, the character development shown in the banters does not feel as "real" as it should.
But their banter give me the impression she doesn't really trust him. She ask him how he knew to approach them and doesn't sound satisfied by his answer (Lucky us, I guess). She asked him how he avoided Templars for so long, if he really lived all along in the woods, etc. She put is story into question.
Being suspicious and seeking answers is part of her duty, yes. It would have been weird if no one had questioned Solas at all. Still, the later banters between the two are much different in tone. Her own inquisitive and honest nature, when not mired in suspicions, is something that Solas would find appealing just as he likes being asked questions by the Inquisitor. When he reassures Cassandra after her personal quest, or when she asks about the effects mortal conflicts have on spirits, it feels like a very real, respectful relationship. As much as Solas' lies allow for that sort of thing, anyway. 